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Kornuta A. A., Lukianenko V. A. Dynamics of solutions of nonlinear functional differential equation of parabolic type. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2022, vol. 30, iss. 2, pp. 132-151. DOI: 10.18500/0869-6632-2022-30-2-132-151

# Dynamics of solutions of nonlinear functional differential equation of parabolic type

Purpose of this work is to study the initial-boundary value problem for a parabolic functional-differential equation in an annular region, which describes the dynamics of phase modulation of a light wave passing through a thin layer of a nonlinear Kerr-type medium in an optical system with a feedback loop, with a rotation transformation (corresponds the involution operator) and the Neumann conditions on the boundary in the class of periodic functions. A more detailed study is made of spatially inhomogeneous stationary solutions bifurcating from a spatially homogeneous stationary solution as a result of a bifurcation of the “fork” type and time-periodic solutions of the “traveling wave” type. Methods. To represent the original equation in the form of nonlinear integral equations, the Green’s function is used. The method of central manifolds is used to prove the theorem on the existence of solutions of the indicated equation in a neighborhood of the bifurcation parameter and to study their asymptotic form. Numerical modeling of spatially inhomogeneous solutions and traveling waves was carried out using the Galerkin method. Results. Integral representations of the considered problem are obtained depending on the form of the linearized operator. Using the method of central manifolds, a theorem on the existence and asymptotic form of solutions of the initial-boundary value problem for a functional-differential equation of parabolic type with an involution operator on an annulus is proved. As a result of numerical modeling based on Galerkin approximations, in the problem under consideration, approximate spatially inhomogeneous stationary solutions and time-periodic solutions of the traveling wave type are constructed. Conclusion. The proposed scheme is applicable not only to involutive rotation operators and Neumann conditions on the boundary of the ring, but also to other boundary conditions and circular domains. The representation of the initial-boundary value problem in the form of nonlinear integral equations of the second kind allows one to more simply find the coefficients of asymptotic expansions, prove existence and uniqueness theorems, and also use a different number of expansion coefficients of the nonlinear component in the right-hand side of the original equation in the neighborhood of the selected solution (for example, stationary). Visualization of the numerical solution confirms the theoretical calculations and shows the possibility of forming complex phase structures.

- Akhmanov SA, Vorontsov MA, Ivanov VY. Generation of structures in optical systems with two-dimensional feedback: Toward the creation of nonlinear-optical analogues of neural networks. In: New Physical Principles of Optical Dada Processing. Moscow: Nauka; 1990. P. 263–325 (in Russian).
- Razgulin AV. The problem of control of a two-dimensional transformation of spatial arguments in a parabolic functional-differential equation. Differential Equations. 2006;42(8):1140–1155. DOI: 10.1134/S001226610608009X.
- Razgulin AV. Nonlinear Models of Optical Synergetics. Moscow: MAKS-Press; 2008. 203 p. (in Russian).
- Akhmanov SA, Vorontsov MA, Ivanov VY, Larichev AV, Zheleznykh NI. Controlling transversewave interactions in nonlinear optics: generation and interaction of spatiotemporal structures. J. Opt. Soc. Am. B. 1992;9(1):78–90. DOI: 10.1364/JOSAB.9.000078.
- Vorontsov MA, Razgulin AV. Properties of global attractor in nonlinear optical system having nonlocal interactions. Photonics and Optoelectronics. 1993;1(2):103–111.
- Chesnokov SS, Rybak AA. Spatiotemporal chaotic behavior of time-delayed nonlinear optical systems. Laser Physics. 2000;10(5):1061–1068.
- Iroshnikov NG, Vorontsov MA. Transverse rotating waves in the non-linear optical system with spatial and temporal delay. In: Walther H, Koroteev N, Scully MO, editors. Frontiers in Nonlinear Optics: The Sergei Akhmanov Memorial Volume. Boca Raton: CRC Press; 1993. P. 261–278.
- Razgulin AV. Finite-dimensional dynamics of distributed optical systems with delayed feedback. Computers and Mathematics with Applications. 2000;40(12):1405–1418. DOI: 10.1016/S0898-1221(00)00249-2.
- Kamenskii GA, Myshkis AD, Skubachevskii AL. The minimum of a quadratic functional, and linear elliptic boundary-value problems with deviating arguments. Russian Mathematical Surveys. 1979;34(3):201–202. DOI: 10.1070/RM1979v034n03ABEH003993.
- Bellman RE, Cooke KL. Differential-Difference Equations. New York: Academic Press; 1963. 465 p.
- Hale JK. Theory of Functional Differential Equations. New-York: Springer-Verlag; 1977. 366 p. DOI: 10.1007/978-1-4612-9892-2.
- Skubachevskii AL. On the Hopf bifurcation for a quasilinear parabolic functional-differential equation. Differential Equations. 1998;34(10):1395–1402.
- Varfolomeev EM. Andronov–Hopf bifurcation for quasi-linear parabolic functional differential equations with transformations of spatial variables. Russian Mathematical Surveys. 2007;62(2):398– 400. DOI: 10.1070/RM2007v062n02ABEH004401.
- Varfolomeev EM. On some properties of elliptic and parabolic functional differential operators arising in nonlinear optics. Journal of Mathematical Sciences. 2008;153(5):649–682. DOI: 10.1007/s10958-008-9141-0.
- Muravnik AB. The Cauchy problem for certain inhomogeneous difference-differential parabolic equations. Mathematical Notes. 2003;74(4):510–519. DOI: 10.1023/A:1026143810717.
- Muravnik AB. Functional differential parabolic equations: integral transformations and qualitative properties of solutions of the Cauchy problem. Journal of Mathematical Sciences. 2016;216(3): 345–496. DOI: 10.1007/s10958-016-2904-0.
- Razgulin AV. Rotational waves in optical system with 2-d feedback. Mathematical Models and Computer Simulations. 1993;5(4):105–119 (in Russian).
- Razgulin AV, Romanenko TE. Rotating waves in parabolic functional differential equations with rotation of spatial argument and time delay. Computational Mathematics and Mathematical Physics. 2013;53(11):1626–1643. DOI: 10.1134/S0965542513110109.
- Romanenko TE. Two-dimensional rotating waves in a functional-differential diffusion equation with rotation of spatial arguments and time delay. Differential Equations. 2014;50(2):264–267. DOI: 10.1134/S0012266114020141.
- Belan EP. On the interaction of traveling waves in a parabolic functional-differential equation. Differential Equations. 2004;40(5):692–702. DOI: 10.1023/B:DIEQ.0000043527.22864.ac.
- Belan EP. On the dynamics of traveling waves in a parabolic equation with the transformation of the shift of the spatial variable. Journal of Mathematical Physics, Analysis, Geometry. 2005;1(1): 3–34 (in Russian).
- Belan EP, Khazova YA. Dynamics of stationary structures in a parabolic problem with reflection spatial variable in the case of a circle. Dynamical Systems. 2014;4(1–2(32)):43–57 (in Russian).
- Belan EP, Shiyan OV. Self-oscillatory combustion regimes along the strip. Dynamical Systems. 2009;(27):3–16 (in Russian).
- Kornuta AA. Metastable structures in a parabolic equation with rotation of the spatial variable. Dynamical Systems. 2014;4(1–2(32)):59–75 (in Russian).
- Belan EP, Lykova OB. Rotating structures in a parabolic functional-differential equation. Differential Equations. 2004;40(10):1419–1430. DOI: 10.1007/s10625-004-0008-y.
- Larichev AV. Dynamic Processes in Nonlinear Optical Systems With Two-Dimensional Feedback. PhD Thesis in Physical and Mathematical Sciences. Moscow: Lomonosov Moscow State University; 1995. 108 p. (in Russian).
- Grigorieva EV, Haken H, Kashchenko SA, Pelster A. Travelling wave dynamics in a nonlinear interferometer with spatial field transformer in feedback. Physica D. 1999;125(1–2):123–141. DOI: 10.1016/S0167-2789(98)00196-1.
- Glyzin SD, Kolesov AY, Rozov NC. Diffusion chaos and its invariant numerical characteristics. Theoretical and Mathematical Physics. 2020;203(1):443–456. DOI: 10.1134/S0040577920040029.
- Karapetiants NK, Samko SG. Equations With Involutive Operators and Their Applications. Rostov-on-Don: Rostov University Publishing; 1988. 187 p. (in Russian).
- Khazova YA, Lukianenko VA. Application of integral methods for the study of the parabolic problem. Izvestiya VUZ. Applied Nonlinear Dynamics. 2019;27(4):85–98 (in Russian). DOI: 10.18500/0869-6632-2019-27-4-85-98.
- Kornuta AA, Lukianenko VA. Functional-differential equations of parabolic type with an involution operator. Dynamical Systems. 2019;9(4):390–409 (in Russian).
- Abramowitz M, Stegun IA, editors. Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. Washington: National Bureau of Standards; 1964. 1082 p.
- Mishchenko EF, Sadovnichiy VA, Kolesov AY, Rozov NK. Autowave Processes in Nonlinear Media With Diffusion. Moscow: Fizmatlit; 2005. 430 p. (in Russian).
- Arecchi FT, Boccaletti S, Ducci S, Pampaloni E, Ramazza PL, Residori S. The liquid crystal light valve with optical feedback: A case study in pattern formation. Journal of Nonlinear Optical Physics & Materials. 2000;9(2):183–204. DOI: 10.1142/S0218863500000170.
- Arnol’d VI, Ilyashenko YS. Ordinary differential equations. In: Encyclopaedia of Mathematical Sciences. Dynamical Systems I. Berlin: Springer-Verlag; 1988. P. 1–148.
- Henry D. Geometric Theory of Semilinear Parabolic Equations. Berlin: Springer-Verlag; 1981. 350 p. DOI: 10.1007/BFb0089647.
- Bateman H, Erdelyi A. Tables of Integral Transforms. Vol. 2. New York: McGraw-Hill Book Company; 1954. 451 p.

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